Are ‘sign’ and ‘sine’ homophones?

At the dinner table last night the conversation wandered into homophones, and at one point ‘sine’ and ‘sign’ were offered as an example, to which I objected. I felt that they are (very subtly) different, but my family and, as it turns out, most of the Internet (including Wiktionary) disagrees with me.

When I say the words, my tongue definitely does distinct motions for either word, though I really can’t be sure whether the result is a different sound.

When I say ‘sine’, the ‘n’ sound is formed by the tip of my tongue pressing against my palate just behind my front teeth. The action and sound are the same as in most ‘-ine’ words like ‘nine’ and ‘fine’.

But when I say ‘sign’ the ‘gn’ is formed by my tongue pressing against the sides and middle-rear of my palate as if I were trying to form a hard ‘g’ sound but the latter never finishes with the vocalized ‘-guh’ typical of hard ‘g’ at the end of a word.

I’m not sure if the resulting sounds are actually different… The only way to tell would be for me to record myself saying a randomized sequence of the two words, replay it, and see if I can actually tell them apart. It would be difficult, though, to prevent myself from over-emphasizing the differences in the sound.